Remark 2.1 (Tapering and banding estimators)  It should be noted that the same rate of convergence can also be attained by tapering and banding estimators with suitable choices of tapering and banding parameters. Specifically,
let \(\hat{{\mathbf {\Sigma }}}^{\rm tp}\)  and \(\hat{{\mathbf {\Sigma }}}^{\rm bd}\)  be respectively the tapering and banded estimators proposed in Cai et al. [1]} and Bickel and Levina [2]} with
\(\hat{{\mathbf {\Sigma }}}^{\rm tp} = \hat{{\mathbf {\Sigma }}}^{\rm tp}_k = (w_{ij}^{\rm tp} \hat{\sigma }_{ij}^\ast )_{1\le i, j\le p} \quad \text{and} \quad \hat{{\mathbf {\Sigma }}}^{\rm bd} = \hat{{\mathbf {\Sigma }}}^{\rm bd}_k = (w_{ij}^{\rm bd} \hat{\sigma }_{ij}^\ast )_{1\le i, j \le p},\) 
